Lesson 2.5 · 10 of 25 lessons

Bases & coordinates

Grow from linear combinations to independence, bases, the fundamental subspaces, and transformations.

Open the interactive animation

01 · Story

What to verify in this scene

Follow the explanations and operations, then identify the mathematical evidence that remains in the final scene.

01
Independent vectors such as b1 and b2 that generate every direction form a basis
02
In this basis, coordinates (2, 1) mean two copies of b1 and one of b2
03
Basis coordinates (2, 1) land at the actual point (1, 3)
Coordinates are coefficients relative to a basis

02 · Notebook

The runnable notebook

This source creates the mathematical objects and controls the order of operations and animation. Read it here, then play the same scene yourself.

2d
zoom 9/10
coordinates on

1, 1  #b1@
-1, 1  #b2@
// Independent vectors such as {{b1}} and {{b2}} that generate every direction form a **basis**
focus b1 b2
checkpoint Bases & coordinates · 1

focus -
sum(b1, b2)  #x@
// In this basis, coordinates `(2, 1)` mean two copies of {{b1}} and one of {{b2}}
checkpoint Bases & coordinates · 2

b1 * 2
// Basis coordinates `(2, 1)` land at the actual point `(1, 3)`\nCoordinates are **coefficients relative to a basis**
focus x

Open the interactive animation

03 · Unit 2

Lessons in this unit

Vector spaces & transformations — Grow from linear combinations to independence, bases, the fundamental subspaces, and transformations.